Minimal models of oriented grassmanians and applications

Mathematica Slovaca

Goutam Mukherjee; Parameswaran Sankaran

Minimal models of oriented Grassmannians and applications

Mathematica Slovaca, Vol. 50 (2000), No. 5, 567--579 Persistent URL:http://dml.cz/dmlcz/136790

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Matli. Slovaca, 50 (2000), No. 5, 567-579 s . ^ l ™ ^ SaSE?..

MINIMAL MODELS OF O R I E N T E D G R A S S M A N N I A N S AND A P P L I C A T I O N S

G O U T A M M U K H E R J E E * — PARAMESWARAN S A N K A R A N * * (Communicated by Julius Korbas)

A B S T R A C T . We construct t h e minimal models for t h e oriented G r a s s m a n n manifold Gn k of all oriented k dimensional vector subspaces of lRn a n d ver- ify t h a t they are formal. As an application we obtain a classification of real flag manifolds according t o nilpotence, which was first established by H. Glover and W . Homer. We also establish a result of K. Varadarajan t h a t t h e classifying space BO(k) is nilpotent if and only if k is odd.

1. Introduction

The purpose of this paper is to give an explicit description of minimal models of oriented Grassmann manifolds. The construction of minimal models of com- pact simply connected homogeneous manifolds is well understood from the work of S u l l i v a n [15] and others. (Cf. [4], [7].) However, we have not been able to find explicit reference for the description, depending only on the parame- ters n and k, 1 < k < n, of minimal model of an oriented Grassmann man- ifold Gn k of oriented k-vector subspaces of Rn. It is our hope that such a description will be useful in answering many questions about Grassmannians (oriented as well as unoriented). Using our description of the minimal model, we prove that Gn k is formal. Of course, this is a well-known result since the oriented Grassmann manifolds are Riemannian symmetric spaces (see [15;

p. 326], [8; p. 158] and [9]). (Cf. Remark 2 below.) We apply our results to show that the action of the fundamental group of Gn k, the Grassmann mani- fold of k planes in Rn, on nk(Gnk) is not nilpotent in case k is even. We deduce a result of H. G l o v e r and W. H o m e r that the real flag mani- fold G(nl,... ,ns) = 0(^/(0^) x ••• x 0(ns)), n = J2 n{ is not nilpo-

l<i<s

tent when one of the ni is even. We also obtain a new proof of a result of

2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : P r i m a r y 55P62, 55P99.

K e y w o r d s : G r a s s m a n n manifold, flag manifold, rational homotopy t h e o r y minimal model, formality, nilpotence.

K. V a r a d a r a j a n that the classifying space BO(k) is nilpotent if and only if k is odd.

We hope that our method of explicit construction and proof, more than the results themselves, will be of some interest. Our approach to nilpotence via the theory of minimal models is probably new.

We now state the main result of this paper. Write k = 2s or 25 + 1 , n — k = 2t or 2t+ 1, 1 < s,t e Z , n > 2k.

Let P denote the polynomial algebra R[pl,... ,ps], where each p• is homo- geneous of degree |p 1=4,7. Define homogeneous elements /i £ P of degree 4j by the equation (1 +px + • • • +ps)(l + hY + • • • + hj + •••) = 1 . Thus hj , j > 1, is a certain polynomial in pl,..., ps.

Introduce elements ok, rn_k of degree k and n — k such that o\ = ps if k is even, ok = 0 if k is odd; Tn_k = 0 if n — k is odd, otherwise it is an indeterminate. Let A be the algebra got by adjoining ok, Tn_k to P. Note that A is a polynomial algebra in even degree generators.

Let M := Mnk denote the commutative differential graded algebra over the reals defined as follows. Recall that commutativity is in the graded sense: for homogeneous elements u and v, uv = ( — l ^ l ^ W .

Case 1:

Let n = 2m, k = 2s, n — k = 2t, s > t. Let M = A[u0,v0,... ,vs_l], \vj\ = 4(£ + j) — 1, 0 < j < s, \u01 = 2ra — 1. The differential d on M is defined as d(A) = 0, d(vj) = ht+j, 1 < j < s, d(v0) = ht - T2n_k, and d(u0) = okTn_k. Case 2a:

Let n = 2ra + 1, fc = 25, n — k = 2t + l, k < ra. Let M = A[vx,..., vs], where

\Vj\ = *(t + J) ~ 1, d(vj) = ht+j, 1 < j < 5, d(A) = 0.

Case 2b:

Let n = 2ra + 1, A; = 2s + 1, n — k = 2t, k < ra. Define M = A[v0,v1,... ,vs], where |^-| = 4(t+j)-l, 0<j<s, d(A) = 0, d(v0) = ht-T2n_k, d(Vj) = ht+j, l<j <s.

Case 3:

Let n = 2ra + 2, A; = 2s + 1, n — k = 2t+ 1. Let M = A[v0,vx, ...,vs], where K-l = 4(t+j)-l, l<j<s, \v0\ = 2ra + l , d(.A) = 0, d(v0) = 0, d(^.) = ht+J, l<j<s.

MAIN THEOREM. Let 2 < k < [n/2]. With notation as above, the commu- tative differential graded algebra Mnk is the minimal model for the oriented Grassmann manifold Gn k .

The minimal model of Gnl = S 'n _ 1, the (n — 1)-sphere, is well known (see [3]). Since Gnk = Gnn_k, the hypothesis that k < [n/2] is not a re- striction.

The paper is organized as follows: In §2 we recall a basic theorem needed in the construction of minimal models of homogeneous spaces. In §3 we prove the Main Theorem stated above and deduce the formality of the oriented Grassmann manifolds. In §4 we obtain results on nilpotence of flag manifolds (Theorem 6) and the classifying space for the orthogonal group (Theorem 7).

2. Minimal Models of homogeneous spaces

Let G be a connected compact simple Lie group. Let H be a closed connected subgroup of G. One has the following description of the minimal model of the smooth homogeneous manifold G/H. Let T C G be a maximal torus in G such that S := T C\ H is a maximal torus of H. Denote by W the Weyl group of G with respect to T and by TV' the Weyl group of H with respect to S.

Let m = d i m T , and let r = dim S. The group W acts on T and hence on the real cohomology algebra H* (BT\ R) of the classifying space of T which is a polynomial algebra over R in m generators each having degree 2. The IV-invariant subalgebra can be identified with the real cohomology algebra of BG. The cohomology algebra H* (BG\ R) is a polynomial algebra R [ FX, . . . , Fm] in homogeneous elements F- having even degrees. (See B o r e l [2].) Similarly, H* (BH-, R) = R [ xx, . . . , xr]. Let p: H* (BG] R) -» H* (BH; R) denote the map induced by the inclusion H C G. Let fi = p(Fi)1 1 < i < m. Now let C = C(G, H) denote the differential graded algebra (d.g.a) H*(BH; R)[ux,..., um], where \u{\ = \f{\ — 1, and the differential d is defined as d(H*(BH]M)) = 0, and d(i^) = f{, 1 < i < m. Note that since \ut\ is odd, graded commutativity implies that u{u- = —u-u^ and, in particular, that u2- = 0.

T H E O R E M 1.

(i) (H. Cartan) With notation as above, the minimal model MG/H of G/H is isomorphic to the minimal model of the d.g.a. ( C ( G , i f ) , d ) .

(ii) (Cf. [15; p. 317, Example (ii), (v)].) The space G/H is formal if for some integer s, 1 < s < m, the sequence f1,...Js is a regular sequence in H*(BH; R), and the elements /5 + 1, . . . , fm belong to the ideal generated by fl, • • •, fs.

A proof is sketched in [16; §4, Chapter 5].

R e m a r k 2. It is known that the sequence f1... , /m is a regular sequence in H* (BH\ R) when H is of maximal rank in G. Hence when H is of maximal rank, G/H is formal. See [16; Chapter 5, Theorem 4.16].

3. Minimal Model of G

In this section we prove the Main Theorem stated in the introduction. We apply Theorem 1 to the case G = SO(n), H = SO(k) x SO(n - k). Write n = 2m or 2m + 1, and k = 2s or 2s + 1, n — k = 2t or 2t + 1, m,s,t being integers. We shall assume that k > 2, since minimal models of spheres are well known. We take T to be the standard maximal torus wThich consists of elements ' = [*i > • • • > U e SO(2m) C SO(n), tj G R, where

t(e{) = cos(27r^.)ei + s i n ^ T r ^ e - ^ ,

*(ci + 1) = " sin(27r^.)e. + cos(27rtj)e.+ 1,

where i = 2j — 1, 1 < i < 2m. (Here the e{ denote the standard basis of Rn .) When n is odd or k is even, H is of maximal rank, m . When n is even and k is odd, S := H D T is of dimension r = s + t = m — 1. For w G W, and t = [ £l 3. . . , £m] G T , w • £ G T is obtained by a permutation of the £. and changing certain of the £ • to — t-. When n = 2m the number of sign changes is to be even. From this it is easy to compute the W-invariant subalgebra of H* (BT; R) = R [ ^ , . . . ,tm], |^. | = 2. Let P. denote the j th elementary symmet- ric polynomial in t\,..., tm, and let vm = t1--tm. Then, if* (BSO(2m)\ R) = R [ PX, . . . , Pm_x, crm], and H* ( £ S O ( 2 m + 1); R) = R [ PX, . . . , Pm]. Note that Pm = am G H*(B5'0(2m);R). The element ( - l )rPr is the Pontrjagin class of the canonical n plane bundle over BSO(n)\ when n = 2m + 1 the (inte- gral) Euler class of the canonical bundle is of order 2 and hence it vanishes in real cohomology (cf. [12]). The calculation of H*(BH;R) is similar. One has H*(BH]R) =R\pl,...,ps,ql,...,qvok,Tn_k], where ak = 0 (resp. a\ = ps) if k is odd (resp. even), and rn_k = 0 (resp. Tn_k = qt) when n — k is odd (resp.

even). The restriction map p: H*(BG;R) -* H*(BH,R) is given by (i) p(Pr) = £ PiQj^'fr* l<r<s + t,

i-\rj—r

(... . . f < V T „ -f c= : 0 if (n,fc) = (2m,2a)>

( H ) / , ( < T-) = j 0 otherwise.

(It is understood that p0 = q0 = 1.)

It can be shown that when (n,k) = (2m, 2s) the elements 6, /1 ?. . . , fm_l

form a regular sequence in the ring R := I7*(M;R). To see this, we note

that, R/(6) is isomorphic to the polynomial ring over R in p

... ,p

, q

,..., q

modulo the ideal generated by p

q

= f

. (This is because 0

= p

q

-) Since, by [3; Proposition 23.7], /-_,..., f

forms a regular sequence in the polynomial algebra R\p

,... ,p

, q

,..., q

] it follows that 0, f

. . . , /

forms a regular sequence in H*(BH;R). Theorem l(ii) shows in particular that the space G

is formal when n and k are both even. Similarly it is seen that /-_,..., f

is a regular sequence in H* (BH; R) when n = 2ra + 1. Tims we conclude that G

is formal when n is odd or k is even. Note that H is of maximal rank in G except when n is even and k odd. Therefore the formality of G

when n is odd or k even follows from Remark 2. In any case, as remarked in the introduction, G

is formal since it is a Riemannian symmetric space. We shall verify formality of G

directly for all values of n and k.

Let P = R\p

,... ,p

] be a polynomial algebra, where \pA = 4j, 1 < / < 5, and let h

G P be defined by (l + /i

+/i

+ -•- + /i

+ -• •) = ( 1 + ^ + - • - + P J "

, where \hA = 4j. We have the following lemma:

LEMMA 3. For any non-negative integer t, the elements h

... >h

form a regular sequence in the polynomial algebra P = R\p

,... , p j .

P r o o f . Let P = R\p

,..., p

]. When 8 = 1, the lemma is obviously true for any t. Assume inductively the statement holds for any t when s is replaced by s - 1.

By the induction hypothesis, for any t, h

,..., h

_

is a regular sequence in P := P/(p

) = % ! , . . . , p

_ J . Equivalent^, h

,..., h

_

,p

is a regular sequence in P . In particular, p

mod (h

,..., h

_

) is not a zero divisor in P/(h

..., -Vs-i) •

t+s +

t+

-iPi + * *' +

tP

= 0 in P . Hence h

= h

p

modulo the ideal (h

,..., ^

) C P .

When t = 0, it is clear that the ideal (h

,..., h

_

) = (p

,... ,p

_-_) and /i

EE ^

p

= p

is clearly not a zero divisor in P/(h

,..., h

_

) in this case. Assume that t > 1 and that the lemma holds when t is replaced by t — 1.

Hence ft

,..., Z^+s^ is a regular sequence. It follows that h

is not a zero divisor in P/(h

,..., /**+,_!>. It follows that h

= h

p

modulo (h

,..., h

_

)

is not a zero divisor in P/(h

,..., / i

) . The lemma follows. • We shall now establish the Main Theorem stated in the introduction.

P r o o f of M a i n T h e o r e m . L e t 2 < f c < [n/2].

Case 1:

Let n = 2ra, k = 2s, n — fc =- 2t, 1 < s < t. In this case the commuta-

tive d.g.a. C

:= C(SO(n), SO(k) x SO(n- k)) is F ( M ; I ) [ w

, . . . ,

] ,

where d(H*(29H;R)) = 0 and du

= f

for 1 < r < ra, d(M

) = 5. Thus,

writing u

= 9 • u

, one has du

= 6 • du

= 6

= p

q

=: /

, and hence

(l + du1 + ... + dum) = (l + f1 + ... + fm) = (l+p1 + ...+p3)(l + ql + --- + qt).

Writing (1 + h1 + ...) = (l+p1...p3)-\ one has hr = _T ff)(-l)'a'pa,

l | a | l = r

where a = ( a _ , . . . , a3) is a sequence of non-negative integers, ||a|| = ___) iai, Ial = _Ca»> ( a ) denotes the multinomial coefficient | a | ! / ( ax! • • •<-*_!), a n d Pa

i

denotes the monomial \\ p?{. One has the following inhomogeneous equation -<«<*

in Cnk: 1 + qx + • • • + qt = (1 + du_ + • • • + d um) ( l + /i_ + • • • ) . In particular we obtain, for 1 < j < s,

ht+j = ~(ht+j-i dui + ""' + hi dut+j-i + dut+j) • When n — k is even, gt = r ^ ^ and hence

/it - Tl_k = - ( / it_ _ du_ + • • • + d ut) .

Let A = R [ P i , . . . , P . - i ^ f c ^ n - f c ] C fT*(B.ff;R). Let A 4M = - 4 [ u0, t ;0, . . .

•••>va-_] denote the commutative d.g.a. over R , where |v.j = | ^t + J| — 1 = 4(t + j) - 1, 0 < j < s, and |u0| = |0| - 1 = 2ra - 1. T h e differential d on Mnk is defined as follows: d(vj) = ht + J. , 1 < j < s , d(v0) = ht — r^_f c, d(u0) =akTn_k, and d(.A) = 0. Clearly Mnk is a free d.g.a. over R .

Note that since t > s, / it, • € A is decomposable for j > 1. Also, since p3 = crk, h3 e A is decomposable. It follows that Mn k is minimal as a d.g.a.

over R . We shall prove that Mn k is a model for the d.g.a. Cnk. From The- orem 1, it will follow that Mnk is a minimal model for Gnk.

Let <\>: Mn k -» Cn k be the _4-algebra homomorphism defined by <f>(u0) = u0, 4>(vj) = -(ht_lJ_1u1+... + utH), l<j<s, <f>(v0) = - ( V1u1 + --- + ut) . Then

<£ is a morphism of d.g.a.'s. Indeed, d(<f>(u0)) = d(u0) = 0 = <f>(0) = <j)(d(u0)), and, for 1 < j < _., one has d(<f>(vj)) = -d(ht_rj_1u1 + .«• + ut + J.) = -(ht+f-i dux + ..-+ d ut + i) = fcl+i = 4>(hi+j) = </>(d(vj)), since d(hr) = 0 as / ir e R [ p _ , . . . , p j C .A. Similarly, d ( # v0) ) = ht - r * , * = ^ ( d ( v0) ) . To show that the chain map <f> induces an isomorphism in cohomology, first observe that H*(Cnk,d) _. H*(Gnk;R). This is because ( Cn i k, d ) is a model for the space Cn k (see Theorem 1). Alternatively, one applies a Koszul complex argument (cf. [11; Chapter XXI, §4]) and uses the fact that # , / _ , . . • , /m_ _ is a regular sequence to see that the cohomology of Cn k is H*(BH\ R ) / ( 0 , / _ , . . . , /m_ _ ) =

H* iPn fc!R) * U s i nS t h e relation ( 1 + / . + - • • + /m) ( l + ' »1+ - ' *) = ( 1 + 9 . + - ' •+<?_), we see that 0 = fr = _T Ptf; , 1 < r < <>i n H*(BH;R)/(<TkTn_k, f^ .. . , /m_ _ )

«+>=r

- II* ( G „i t; R) • In particular, qj = hj for any j , l<j<t and r*_fc = qt = ht

in H*(BH;R)/(akTn_k,fv , / J . Hence

-**(G„,

;R) =%!,••• , P , - i , ' t

, v J / ( V i V . - i . V

- ^ - "T

) •

Using Lemma 3, one sees that Tn_k, ht — Tn_k,ht+1,..., ht+8_1 and crk,ht — Tn_k,ht+1,..., ht+s_1 are regular sequences in A. From [3; Lemma 23.6] it fol- lows that 6,Tn_k—ht,ht+1,...,hm_1 is a regular sequence in A. Again by ap- plying a Koszul complex argument we obtain that

H*(Mn<k,d) e. A/(Tl_k - hvht+1,...,hm_x,crkTn_k) 9. H*(Gny,R) . Under our identifications, the map (j) actually induces the identity map of H*(Gn k,R) . This proves that M k is quasi isomorphic to Cnk and hence it is the minimal model of G k in this case.

Case 2:

Let n = 2m + 1 = 2s + 2t + 1, k = 2s, or 25 + 1. We assume that fc < m (equivalently s < t with equality only if fc = 2s). In this case Cnk =

H*(BH;R)[u

,...,u

], where d(H*(BH;R)) =0, duj-fj, 1 < j < m.'

Let A C H*(BH;R) be the polynomial algebra over R in generators P i r - . , PM, ^ (resp. P i , . . . , P5, Tn_ f e ) for fc even (resp.fc odd). Thus ps = ^ when fc = 25.

Subcase (a):

Let fc = 25. Let A tn fc = A ^ , . . . ,vs] be the d.g.a. over R, where 1^1 =

| / ij + t| - l = 4 ( j + t ) - l and d ( ^ ) = ht+j, l<j<s, d(A) = 0. The free d.g.a.

Mnk is minimal since the ht+J- are decomposable for j > 1. The A-algebra

map (j) : Mnk -+ Cnk defined by <f>(Vj) = -(hj^^u^ + • • • + uj+t), l<j<s, is a morphism of d.g.a.'s. Also, the elements ht+j, 1 < j < s, form a regular

sequence in A. Therefore arguing as in case 1 above, we conclude that (j) is a quasi isomorphism. Hence Mnk is a minimal model of Gn k.

Subcase (b):

Let fc = 25 + 1 . Let Mnk = -4[t>0,i'1,... ,-uJ be the commutative d.g.a. over R, where | ^ | = \ht+j\ - l = 4(t + j) - 1, 0 < j < s, and d(v0) = ht - T*_k, d(vA = ht+j, 1 < j < 5, and d(A) = 0. Since s < t, ht+- is decomposable for j > 0. Therefore Mnk is minimal. The A-algcbra map 6: Mnk —> Cn k

defined by ^(Vj) = —(ht+J_lul + • • • + Uj), 0 < j < s, is a morphism of d.g.a.

over R. Using the fact that ht — T^_k, hi+1,...,hm is a regular sequence in A = R\p1,... ,Ps,Tn_k], we conclude, as before, that Mn h is a minimal model

Case 3:

Let n = 2ra + 2, fc = 2$ + l , n — fc = 2£ + l , 1 < 5 < t . In this case the subgroup SO(k) x SO(n — k) is not of maximal rank in SO(n). The c.d.g.a.

Cnk has the description H*(JBH;E)[u1,... , um, i i0] with | i * | = A(t + j ) — 1,

|ii01 = 2ra + 1 = n — 1, and du- = / •, 1 < j < ra, and du0 = 0.

Let A = P = % ! , . . . , ps] C H*(BH;R). Let yVfnfc denote the d.g.a.

A ^ , . . . , ^ , ^ ] , where \vj\ = A(t + j ) - 1, 1 < j < ' 5 , |u0| = 2ra + 1, d ( f ) = ^J + t, 1 < j < «s, df0 = 0, and d(A) = 0. As before, Mnk is free and minimal.

The A-algebra map </>: Mn k —r Cn k defined by 4>(v-) = u - , 0 < j < 5, is a chain map as can be verified as in Case 1. Note that the cohomology of Mn k can again be computed using the Koszul complex of A with respect to the sequence / it + 1, . , . , / it + s, 0 G A. Again using the fact that ht+1,... ,ht+s is a regular sequence, a simple calculation leads to:

H*(Mny,R) = A[u0}/(ht+1,...,ht+s).

This is also the cohomology of Cn k and as in Case 1, we see that 0 induces isomorphism in cohomology. Hence Mnk is the minimal model of Gn k.

This completes the proof of the Main Theorem. D COROLLARY 4 . The oriented Grassmannian Gn k is formal for all 1 < k < n.

In particular all Massey products in H* (Gn k] R) vanish.

P r o o f . First let n = 25 + 2t + 2, k = 25 + 1. With notation as above, note that the A-algebra map Mnk -> A[u0]/(ht+1,...,ht+s) = H*(Mny,R) defined by v0 i-> u0, v • i-> 0, 1 < j < s, is a map of d.g.a.'s where the differential on H*(Mn k]R) is defined to be zero. Hence Gnk is formal.

The same argument as above shows that Gn k is formal for all parities of n

and k. • COROLLARY 5. Let dimRv7rr(<5n^) cg)z R) = 7 rr.

(i) Let n = 2s + 2t, k = 2s} l<s<t.

Then £ 7Trzr = 1 + zn~l + z2s + z2t + zAt~l + £ (z^ + z4^ ' ) "1) .

r > 0 l < j < 5

(ii)

(a) Let n = 25 + 2t + 1, fc = 25, s<t.

Tfen E v r = 1 + ^ + ^4 ( s + t H+ £ (z*i + z4^ -1) .

r > 0 l < j < 5

(b) Let n = 2s + 2£ + 1. k = 25 + 1, s <t.

Then £ frrzr = 1 + z2t + zU~l + £ (*4j + £4 ( i + j ) _ 1) .

r > 0 l < j < s

(iii) Let n = 2s + 2t + 2. fc = 25 + 1, 1 < 5 < t.

Then £ nrzr = 1 + zn~x + £ (z4^' + z4^ ' ) -1) .

r > 0 l < j < 5

_ P r o o f . This follows from the above description of the minimal model of Gnk and the fact that Homz(7rr(Gn k),R) is isomorphic to the r t h degree component of the graded vector space Mnk/V, where V denotes the ideal

Mnk-Mnk of "decomposable elements". •

4. Nilpotence of Grassmannians and related spaces

Let X be a path connected topological space with base point x. Recall that A" is called nilpotent if the fundamental group 7r = 7r1 (Ar, x) of Ar is nilpotent as a group and all the higher homotopy groups of X are nilpotent as modules over the integral group ring Zrr. That is, denoting the augmentation ideal of ZTT by 7, Ar is nilpotent if and only if n is nilpotent and, for each n > 2, there exists an integer N = N(n) such that IN • 7rn(A, x) = 0.

A path connected topological space Ar is said to be of finite Q-type if Hn(X]Q) is finite dimensional for all n > 1. If X is nilpotent, then X is of finite Q type if and only if HX(X; Q) and nn(X) ® Q are finite dimensional for all n > 2. When X is a nilpotent space of finite Q-type, one associates to Ar the minimal model Mx of the Sullivan-de Rham complex of X which is a c.d.g.a. over Q. The minimal model Mx contains all the rational homotopy information of X in this case. That is, Mx and MY are quasi isomorphic if and only if Ar and Y are of same rational homotopy type, where both X and Y are of finite Q-type. We refer the reader to [1; Chapter 2] for details. (See also [4; §9].) In case Ar is a smooth manifold, it is more convenient to work with the real homotopy theory via the minimal model of the de Rham complex of X.

Let n j , . . . , ns be a sequence of positive integers, and let n = ^2 ni • Denote

l<i<s

by Ar = G(nx,..., ns) the flag manifold consisting of flags ( V1 5. . . , Vs), where

\] is an n{ dimensional vector subspace of Rn such that V{ ± V- if i ^ j , and \\ 0 • • • © Vs = Rn . The flag manifold X can be identified with the coset space 0(n)/(0(nx) x ••• x 0(ns)) so that it is naturally a smooth compact manifold of dimension ^ n{n •. When s = 2, it is identified with the Grass-

l<i<j<s

mannian Gnni. The universal covering of the flag manifold is the oriented flag manifold Ar = G(nx,... , n j = SO(n)/'(50(71,) x • • • x SO(ns)), which consists of "oriented flags", that is, flags ( V1 ?. . . , Vs) together with orientations on each vector space V{ so that the direct sum orientation on ^ V{ = lRn coincides with the standard orientation on En ._The natural map p: X -> X that forgets the orientations on oriented flags of Ar is the covering projection. The deck transfor- mation group is generated by the involutions a •, 1 < i < s, which reverse the orientation on zth and 5 th vector space in each oriented flag, and is isomorphic

to ( Z / 2 )5 - 1. We note that, when n{ and ns are odd, a{ can be realized as multiplication on the left by the element A{ € SO(n), where A{ has, as a block diagonal matrix j th block down the diagonal the identity matrix of size n • if j ^ i,s and the zth and 5th block being negative identity matrices of sizes n{

and ns respectively.

In this section we prove the following theorem:

THEOREM 6. ( G l o v e r — H o m e r [6]) Let X = G(nx,... ,n3), s > 2, de- note the real flag manifold. Then the following are equivalent:

(i) all the n{ are odd, (ii) X is simple, (iii) X is nilpotent.

THEOREM 7. (Varadarajan) The classifying space BO(k) is nilpotent if and only if k is odd.

We need the following lemma (cf. [14; Chapter 7, §3, Lemma 7]). We shall denote the free homotopy classes from the n-sphere to Y by 7rn(F). Note that when Y is a simply connected space, nn{Y) may be identified with 7rn(F, y ) . LEMMA 8. Let p: X —•> X be the universal covering projection of a path con- nected finite CW complex, and let r > 2 . Then p induces an isomorphism between TT (X) and 7rr(X,x), which is compatible with the action of the deck transformation group on nr(X) and that of TTX(X,X) on irr(X,x).

P r o o f of T h e o r e m 6.

(i) -==> (ii): Assume that all the n{ are odd. Since SO(n) is connected, maps induced on X by multiplication by elements of SO(n) are all homotopic to the identity map. In particular the at, 1 < i < s, are homotopic to the identity map of X. Since the a • generate the deck transformation group of the universal covering projection X —> X, it follows from Lemma 8 that the action of n1(X) = ( Z / 2 )s _ 1 on any 7rr(X) is trivial and so the space X is simple-

It is evident that (ii) implies (iii).

(iii) =-> (i): We will first assume that 5 = 2 so that X is the Grassmann manifold X = Gnn . For simplicity of notation let k = nx. Assume that at least one of the integers k,n—k is even. We will prove that X is not nilpotent.

Indeed, let r be an even integer in the set {k,n — k}. We will prove that the action of the generator of TT1(X) = Z / 2 on 7Tr(X)®zR has —1 as an eigenvalue.

When r = n — k > k it will be convenient to denote by ar the element that was denoted Tn_k in §3.

We first note that the deck transformation a := ax of the covering projection Gn k —r Gn k is not homotopic to the identity. In fact a reverses the orientation

on the canonical k-plane bundle over X = G , . When r is even, the Euler class ar of the canonical oriented r -plane bundle is not torsion and in real cohomology one has a*(ar) = —or. The map a induces an endomorphism a of the d.g.a. Mn k which is unique up to chain homotopy. The map a induces an involution [a], which depends only on the chain homotopy class of d , of the graded R-vector space MnkjV such that the following diagram commutes, where the vertical arrows are natural isomorphisms (cf. [5]). Here a* denotes the R-linear involution on Homz(7rr(Gn fc),R) induced by a.

(M„,JVY -!=L> (M„

JVY

I I

Homz(7rr(Gnife),IR) - ^ - > Homz(7rr(Gn > f e),E)

Note that by Corollary 5 the vector space Homz(7rr(Gn fc),R) is non-zero. In fact the class crr is not in the ideal V of decomposable elements of Mn k. Also [a](crr) = — crr since in de Rham cohomology a* maps the Euler class of the canonical r-plane bundle to its negative. Hence it follows that the —1 eigenspace of a*: Homz(7rr(Gn A;),R) —> Homz(7rr(Gn fc),R) is non-zero. It follows that

— 1 is an eigenvalue for the action of the generator of the fundamental group of

Gn,k o n *AGn,k) ® zR- H e n c e Gn,k i s n o t nilpotent.

Now let s > 3 and assume without loss of generality that nx is even.

One has a natural inclusion j : Y := G(nx,n2) —r X and a projection map q: X -> G{n^n2 + ••• + n8) =: Z. Explicitly, j(A) = (A,E2,...,E8) and q(\\,..., Vs) = Vl, where Er denotes the span of the standard basis vectors et, nx-\ h nr_x + 1 <t <nx-\ \-nr. Note that qoj is the natural inclusion of Y into Z induced by the inclusion of Rn- +n- into Rn and hence induces an isomorphism of fundamental groups. Denote by f:Y—>Z a lift of q o j o p to Z := Gn n i + n 2, where p: Y := Gn i + n 2 ? n i -> Y is the universal covering projection. The map / pulls back the canonical nx -plane bundle on Z to that on Y. Hence it maps the Euler class an (Z) to the class +an (Y). Replacing / by / o a if necessary one may as well assume that / * (an (Z)) = ani (Y). As in the case when 5 = 2, using naturality properties of minimal models, one con- cludes that the morphism of d.g.a. induced by / maps the class a (Z) G Mg to O~n (Y) 6 My • This implies that the —1 eigenspace for the action of n^Y) (via the isomorphism of fundamental groups induced by qoj) on Hom(7rn (Z),R) is non-zero. Hence the action of 7r1 (Y) via the monomorphism of fundamental groups induced by j on Hom(7rn (X),R) must have non-zero - 1 eigenspace.

This clearly implies that the action of ^X(X) on 7r (X) is not nilpotent. •

R e m a r k 9. Theorem 7.2 of [10] can be readily applied to show that (iii) = > (i) once this is known for Grassmannians. Our proof, however, exhibits a higher homotopy group of X on which the action of the fundamental group is not nilpotent. Alternatively, one can deduce the same result from our result for Grassmannians using the second Claim in the proof of [10; Theorem 7.2]. We record, as a corollary of the the above proof, the following proposition for pos- sible future reference.

PROPOSITION 10. Let X = G(nv...,ns), s > 2 . If n{ is an even integer, then itn.(X) is not nilpotent as a 7r1(X) module.

P r o o f of T h e o r e m 7. Note that BO(k) — |J Gn k, where we regard

n>2k

Gn k as the subspace of Gn + 1 k in the usual way, considering the vector space Rn as the subspace of Mn + 1 consisting of those vectors with last coordinate being zero. The inclusion map in: Gn k —» BO(k) is an (n — k) -equivalence and hence, given any r > l , f o r n > k + r + l , the map in induces isomorphism of the r th homotopy groups. Also the action of the fundamental group of Gn k on nr(Gn k,x) is compatible with the action of the fundamental group of BO(k) on 7rr(BO(k), x) via the map induced by in. In particular if k is even, choosing r — k and n > 2k + 1, one sees from our proof of Theorem 6 that BO(k) is not nilpotent.

When k is odd, one can always choose n to be even (in addition to n >

k + r + 1) to conclude that the fundamental group of BO(k) acts trivially on nr(Gn k) for any r > 1. Hence we conclude that the space BO(k) is nilpotent.

In fact we have shown that BO(k) is simple. D

Acknowledgement

We thank K. Varadarajan for his interest in this problem and for making available to us his paper [17]. We thank J. Korbas for his comments on an earlier version of this paper and, in particular, for pointing out to us the work of H. G l o v e r and W. H o m e r [6].

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Received December 1, 1998 * Stat-Math Unit

Revised Anril 15 1999 Indian Statistical Institute 203 Barrackpore Trunk Road Calcutta 700 035

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** Chennai Mathematical Institute 92 G.N. Chetty Road

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E-mail: goutam@isical.ac.in sankaran@smi.eгnet.in